Two vertices of a triangle are (0, 2) and (4, 3). If its orthocentre is at the origin, then its third vertex lies in which quadrant?
- A.Fourth
- B.Third
- C.Second
- D.First
Show answer and solution
Correct option: C
Solution
Let the third vertex be (h, k). The orthocenter is at origin (0,0).
The altitude from (h,k) is perpendicular to the side joining (0,2) and (4,3). Slope of this side = (3-2)/(4-0) = 1/4. So the altitude from (h,k) passes through origin with slope -4: k/h = -4, giving k = -4h.
The altitude from (0,2) is perpendicular to the side joining (h,k) and (4,3). Slope of that side = (k-3)/(h-4). This altitude passes through (0,2) and origin, so its slope = 2/0... wait, altitude from (0,2) passes through origin means the line from origin to (0,2) is vertical (x=0). So this altitude is the y-axis (x=0), which means the side joining (h,k) and (4,3) must be horizontal (slope 0): k = 3.
From k = -4h and k = 3: h = -3/4.
Third vertex is (-3/4, 3), which has negative x and positive y → Second quadrant.
The altitude from (h,k) is perpendicular to the side joining (0,2) and (4,3). Slope of this side = (3-2)/(4-0) = 1/4. So the altitude from (h,k) passes through origin with slope -4: k/h = -4, giving k = -4h.
The altitude from (0,2) is perpendicular to the side joining (h,k) and (4,3). Slope of that side = (k-3)/(h-4). This altitude passes through (0,2) and origin, so its slope = 2/0... wait, altitude from (0,2) passes through origin means the line from origin to (0,2) is vertical (x=0). So this altitude is the y-axis (x=0), which means the side joining (h,k) and (4,3) must be horizontal (slope 0): k = 3.
From k = -4h and k = 3: h = -3/4.
Third vertex is (-3/4, 3), which has negative x and positive y → Second quadrant.
[JEE Main 2019 · 10 Jan · Shift II]